On fixed point theorems and social choice paradoxes (Q908824)

In the case of two voters and two-dimensional choice spaces the author shows that the nonexistence of a continuous and anonymous rule that satisfies unanimity is equivalent to a fixed point problem. Specifically, the main result is that the existence of such a choice rule is equivalent to the existence of a continuous map from the closed unit disk of \({\mathbb{R}}^ 2\) into itslf without fixed points. The author also indicates how the result can be extended to higher-dimensional choice spaces and any finite number of voters.